Miter Angles For Common Polygons

Every regular flat-frame polygon — a box, a picture frame, a planter rim, a segmented-ring layer — needs the same one-line arithmetic to find its miter angle, and once the derivation is fixed in your head, you never need to look up a table again.

Where the formula comes from

Walk around the outside of any regular N-sided polygon and you turn through a full 360° by the time you're back where you started, split evenly across N corners — so each corner turns the path by 360/N degrees. At that corner, two boards meet and each contributes exactly half of that turn, because the miter cut on each board bisects the angle between the two faces meeting there. That gives the whole rule in one line:

miter angle (per end) = 180 / N

This is a flat-frame formula — it assumes the sides stand straight up with no lean or flare. A box, frame, or ring whose sides tilt out from vertical needs a second angle (a blade-tilt, or bevel) on top of this one; that's compound work, covered separately.

The table, worked from the formula rather than memorized

Sides (N)	Miter angle
3	60°
4	45°
5	36°
6	30°
7	25.71°
8	22.5°
9	20°
10	18°
11	16.36°
12	15°

Notice the angle shrinks as the side count climbs — more corners means each one turns the path less, so the cut gets shallower, not steeper. A 3-sided shape needs the most aggressive cut of the common set (60°) and a 12-sided one the gentlest (15°); anything with more sides than that keeps following the same 180/N pattern, just with smaller and smaller per-cut angles.

Why 5, 7, 9 and 11 sides are the ones that go wrong

Even-numbered, "nice" counts — 4, 6, 8, 10, 12 — land on angles most people can estimate close enough by eye to catch a gross error: 45°, 30°, 22.5°, and so on are familiar marks on a miter gauge. The odd counts don't cooperate the same way. Five sides gives 36°, not the 45° or 40° a quick mental guess tends to land on. Seven sides gives 25.71°, a repeating decimal that cannot be dialed in exactly on a saw with only whole-degree markings — a digital angle gauge or a shop-made jig is the honest way to hit it, not eyeballing the printed scale. This is the single most common source of an out-of-square odd-sided box: not a math error, but a saw that physically can't be set to the precise fractional angle the math calls for, and a builder who rounds without realizing the rounding compounds around the loop.

Error compounds around the ring, not just at one joint

A flat frame or ring closes on itself, so a small miter error at one corner doesn't stay local — it propagates around every remaining joint and shows up as a gap at the LAST seam, the one place you can't sneak a shim in without it being obvious. The fewer the sides, the worse a given error looks: on a 3-sided frame, a half-degree mistake gets amplified across only two other corners before the loop has to close, leaving a visible wedge-shaped gap; on a 12-sided ring, the same half-degree error is spread thinner across eleven other joints and is far more forgiving. Cutting one extra test corner from scrap stock and checking it against a known square or protractor before committing the real material is cheap insurance, and it matters more the fewer sides the project has.

Using this with the calculator

Run any side count through the angle calculator to get the exact figure to as many decimal places as your saw's digital readout supports, rather than rounding the table above by hand. For a specific side count's fast lookup, including the ones covered here, see the individual miter-angle answer pages linked from this site's answers index — this page is the one place the full derivation and the complete table live; the individual pages are quick single-number references back to it.